## Why I love working with my kids

Saw this problem yesterday via a Fawn Nguyen tweet:

Thought it would be a fun problem to talk through with the boys, but then my older son had to run an errand with wife so it ended up just being my younger son.

His approach just blew me away. I’m so glad to have the opportunity to have conversations like this with my kids:

[update from the day after originally publishing follows]

This morning I went through the question with my older son. He struggled with the problem much more than my younger son did. One of the contributing factors to his struggle is that he chose to keep track of the number of leaves on the ground rather than the number that fell each minute.

It was interesting to watch him work through the difficulties. He looked for lots of different patterns before finding the right one. I broke up his solution into 3 parts:

After he finished we had a neat discussion about how to add up the series:

$1 + 2^1 + 2^2 + \ldots + 2^n$

He had a neat approach, which was essentially mathematical induction. You know that 1 + 2 is 4 – 1. That helps you see that 1 + 2 + 4 is 8 – 1, and so on. That was a neat way to think about the sum and led to a fun (and basic) conversation about mathematical induction.

Thanks to the Math Forum for this fun problem.

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### Comments

2 Comments so far.
1. Some possible extensions:
(1) how many leaves are on the ground at the end of the 12th minute? The 24th?
(2) how many leaves are on a tree (clearly an ambiguous question)?
(3) how much would the leaves that fall in the 12th minute weigh?
(4) when would the leaves on the ground be equal to the mass of the earth?
(5) is this a good model for describing leaves falling from a tree?
(6) (presuming an answer to question 4) what is a better model?

2. BTW, a similar growth pattern comes up in Towers of Hanoi (which I was mistakenly re-solving b/c I misread the Reve’s Puzzle from Dudenay’s Canterbury Puzzles).